Here, we set up a standard benchmark for atmospheric models
in a spherical configuration, used to compare dynamical cores within
the 2016 edition of DCMIP (Dynamical Core Intercomparison Project) [1]},
following the prescription from [2]}.
The test case is designed to idealize global weather evolution in the midlatitudes.
The model is initialized with a deep-atmosphere, balanced steady-state
solution that is axisymmetric about the Earth’s rotation axis,
using analytical formulae from [2]}.
The planetary rotation vector is \(\mathbf {\omega } = (0, 0, 2\Omega )\) 
with \(\Omega = 7.29212 \times 10^{-5} \frac{1}{\text{s}}\)  .
A localized Gaussian hill perturbation of the
zonal wind field is introduced in the northern midlatitudes, which triggers
the evolution of a baroclinic wave over the course of several
days.
In the initial 7-day period, the wave dynamics is linear, after which the
dynamics transitions to a nonlinear regime with a steepening and breaking of the wave.
To compare with other models, in our presentation the focus is on the first
15 days, and particularly day 8 (before the wave breaking) and day 10 (after the wave breaking).
However, to properly evaluate the robustness of the entropy-stable scheme, we confirmed
that all our simulations successfully run for over 100 days without any explicit
dissipation.
To resolve the sphere, we use an equiangular cubed-sphere mesh with \(K_h\)  elements per horizontal side
on each of the six panels and \(K_v\)  elements in the vertical.
The model top is 30 km and rigid lid boundary conditions are assumed at the top and
bottom.
We perform two simulations with polynomial orders 3 and 7. The simulation
with polynomial order 3 uses \(K_h = 30\)  and \(K_v = 8\) ,
while the simulation using polynomial order 7 uses \(K_h = 15\)  and \(K_v = 4\) .
These choices ensure that the total number of degree of freedom is the same for both simulations,
corresponding to about 100 km average horizontal resolution and 32 vertical levels.
Time stepping was performed using the second order additive Runga Kutta scheme
from [4]} with CFL = 3.0.
<FIGURE><FIGURE>